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A Unified Approach to Analysis and Design of Denoising Markov Models

Yinuo Ren, Grant M. Rotskoff, Lexing Ying

TL;DR

This paper establishes a unified, generator-based framework for denoising Markov models that encompasses diffusion, jump, and general Lévy-type forward dynamics. By tying the backward process to a generalized Doob's $h$-transform, it derives explicit backward generators and a variational objective that directly minimizes the measure-transport discrepancy via KL divergence, while extending score-matching to broad dynamics. The approach unifies continuous and discrete diffusion formulations and provides a practical meta-algorithm for training and inference, with explicit losses that generalize classical diffusion score matching. The authors validate the framework through novel forward dynamics, including geometric Brownian motion and jump processes, demonstrating flexibility and potential for modeling complex distributions beyond standard diffusion setups.

Abstract

Probabilistic generative models based on measure transport, such as diffusion and flow-based models, are often formulated in the language of Markovian stochastic dynamics, where the choice of the underlying process impacts both algorithmic design choices and theoretical analysis. In this paper, we aim to establish a rigorous mathematical foundation for denoising Markov models, a broad class of generative models that postulate a forward process transitioning from the target distribution to a simple, easy-to-sample distribution, alongside a backward process particularly constructed to enable efficient sampling in the reverse direction. Leveraging deep connections with nonequilibrium statistical mechanics and generalized Doob's $h$-transform, we propose a minimal set of assumptions that ensure: (1) explicit construction of the backward generator, (2) a unified variational objective directly minimizing the measure transport discrepancy, and (3) adaptations of the classical score-matching approach across diverse dynamics. Our framework unifies existing formulations of continuous and discrete diffusion models, identifies the most general form of denoising Markov models under certain regularity assumptions on forward generators, and provides a systematic recipe for designing denoising Markov models driven by arbitrary Lévy-type processes. We illustrate the versatility and practical effectiveness of our approach through novel denoising Markov models employing geometric Brownian motion and jump processes as forward dynamics, highlighting the framework's potential flexibility and capability in modeling complex distributions.

A Unified Approach to Analysis and Design of Denoising Markov Models

TL;DR

This paper establishes a unified, generator-based framework for denoising Markov models that encompasses diffusion, jump, and general Lévy-type forward dynamics. By tying the backward process to a generalized Doob's -transform, it derives explicit backward generators and a variational objective that directly minimizes the measure-transport discrepancy via KL divergence, while extending score-matching to broad dynamics. The approach unifies continuous and discrete diffusion formulations and provides a practical meta-algorithm for training and inference, with explicit losses that generalize classical diffusion score matching. The authors validate the framework through novel forward dynamics, including geometric Brownian motion and jump processes, demonstrating flexibility and potential for modeling complex distributions beyond standard diffusion setups.

Abstract

Probabilistic generative models based on measure transport, such as diffusion and flow-based models, are often formulated in the language of Markovian stochastic dynamics, where the choice of the underlying process impacts both algorithmic design choices and theoretical analysis. In this paper, we aim to establish a rigorous mathematical foundation for denoising Markov models, a broad class of generative models that postulate a forward process transitioning from the target distribution to a simple, easy-to-sample distribution, alongside a backward process particularly constructed to enable efficient sampling in the reverse direction. Leveraging deep connections with nonequilibrium statistical mechanics and generalized Doob's -transform, we propose a minimal set of assumptions that ensure: (1) explicit construction of the backward generator, (2) a unified variational objective directly minimizing the measure transport discrepancy, and (3) adaptations of the classical score-matching approach across diverse dynamics. Our framework unifies existing formulations of continuous and discrete diffusion models, identifies the most general form of denoising Markov models under certain regularity assumptions on forward generators, and provides a systematic recipe for designing denoising Markov models driven by arbitrary Lévy-type processes. We illustrate the versatility and practical effectiveness of our approach through novel denoising Markov models employing geometric Brownian motion and jump processes as forward dynamics, highlighting the framework's potential flexibility and capability in modeling complex distributions.

Paper Structure

This paper contains 52 sections, 23 theorems, 198 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Problem Setting and Notations
  5. (Continuous) Diffusion Models
  6. Discrete Diffusion Model
  7. Denoising Markov Models
  8. Mathematical Background
  9. Time-Reversal of Forward Process
  10. Change of Measure
  11. Meta-Algorithm
  12. Training Process.
  13. Inference Process.
  14. Examples
  15. Diffusion Process
  16. ...and 37 more sections

Key Result

Theorem 3.2

Under certain regularity assumptions on the densities $p_t$ (ass:density), the backward generator ${ \hbox{{\cr \hidewidth\reflectbox{$\m@th\vec{}\mkern4mu$}\hidewidth\cr {} $\m@th{\mathcal{L}}$\cr }}}_t$ can be explicitly expressed in terms of the forward generator ${\mathcal{L}}_t$ and the densit where ${\mathcal{L}}_t^*$ is the adjoint operator of ${\mathcal{L}}_t$ (def:adjoint), and is the c

Figures (4)

  • Figure 1: Diagram of our framework of the denoising Markov model. Quantities and relations related to the true forward and backward process are colored in red, while those estimated are colored in blue. $\Gamma_t^*$ and ${ \hbox{{\cr \hidewidth\reflectbox{$\m@th\vec{}\mkern4mu$}\hidewidth\cr {} $\m@th\Gamma$\cr }}}_t$ are the carré du champ operator associated with the adjoint generator ${\mathcal{L}}_t^*$ and the backward generator ${ \hbox{{\cr \hidewidth\reflectbox{$\m@th\vec{}\mkern4mu$}\hidewidth\cr {} $\m@th{\mathcal{L}}$\cr }}}_t$, and the ratio $\eta_t = \varphi_t p_t^{-1}$ is colored in purple.
  • Figure 2: Empirical results of denoising Markov models using geometric Brownian motion as the forward process. The generated samples from $\widehat{q}_T$ (in blue) closely match the target distribution $p_0$ (dashed line in 1-D and contour lines in 2-D in red), highlighting the effectiveness of the learned backward process.
  • Figure 3: Visualization of generated backward trajectories using geometric Brownian motion as the forward process. Samples generated via the learned backward dynamics (in blue) evolve from the initial distribution $q_0$ with the estimated score function $\widehat{{\bm{s}}}_t({\bm{x}})$ into the target distribution $p_0$ (in red), further validating the theoretical robustness and practical effectiveness of our denoising Markov models.
  • Figure 4: Visualization of generated backward trajectories using the jump process as the forward process. Samples from $q_t$ generated via the learned backward dynamics (in blue) are compared with the corresponding distribution $p_{T-t}$ from the forward process (in red). Results for three distributions (Chessboard, Swiss roll, Moons) demonstrate close agreement between the generated and target distributions.

Theorems & Definitions (43)

  • Definition 3.1: Forward Generator, Informal Version of \ref{['def:right_and_left_generator']}
  • Theorem 3.2: Time-Reversal of Forward Process, Informal Version of \ref{['thm:time_reversal_app']}
  • Lemma 3.4: Informal Version of \ref{['prop:K_cev_L']}
  • Theorem 3.5: Generalized Doob's $h$-Transform chetrite2015nonequilibrium
  • Theorem 3.6: Change of Measure, Informal Version of \ref{['cor:change_of_measure']}
  • Corollary 3.7
  • Corollary 3.8: Score-Matching
  • Remark 3.9: Comparison with benton2024denoising
  • Theorem 3.11: Meta-Error Bound for Denoising Markov Models
  • Example 4.1: Continuous Diffusion Model
  • ...and 33 more